Final answer:
To find the point at which the marginal cost equals the average cost, set the two cost functions equal to each other and solve for q. In this case, there is no point at which the marginal cost equals the average cost.
Step-by-step explanation:
To find the point at which the marginal cost equals the average cost, we need to set the two cost functions equal to each other and solve for q.
Marginal cost (MC) is the derivative of the average cost (AC) function. So, we can set the derivative of AC equal to MC and solve for q:
AC(q) = MC(q)
0.3q^2 - 0.5q + 13 = 0.3q - 0.5
0.3q^2 - 0.8q + 13.5 = 0
We can solve this quadratic equation to find the value(s) of q at which MC equals AC using the quadratic formula:
q = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values from our equation:
q = (-(-0.8) ± √((-0.8)^2 - 4(0.3)(13.5))) / (2(0.3))
q = (0.8 ± √(0.64 - 16.2)) / 0.6
q = (0.8 ± √(-15.56)) / 0.6
The discriminant (-15.56) is negative, which means there are no real solutions for q that satisfy the equation. Therefore, there is no point at which the marginal cost equals the average cost.